This short note is a commentary on a 2024 article by Mathur and Shpitser in the Journal, with the aim to enlarge the class of graphs for which the conditional average treatment effect is nonparametrically identified, by allowing the outcome to be on the pathway between the treatment and the selection indicator. A first straightforward generalization is possible when (1) the outcome $Y$ is binary, and (2) the population prevalence of $Y$ is known a priori or can be made the object of a sensitivity analysis. Furthermore, identification of the effect is possible also for $Y$ having any nature, provided that a selection bias breaking node $V$ exists and the population prevalence of $V$ is known.

In this paper, we assume that cause–effect relationships between random variables can be represented by a Gaussian linear structural equation model and the corresponding directed acyclic graph. Then, we consider a situation where a set of random variables that satisfies the front-door criterion is observed to estimate a total effect. In this situation, when the ordinary least squares method is utilized to estimate the total effect, we formulate the unbiased estimator of the causal effect on the variance of the outcome variable. In addition, we provide the exact variance formula of the proposed unbiased estimator.

This paper assumes a context in which cause–effect relationships between random variables can be represented by a Gaussian linear structural equation model and the corresponding directed acyclic graph. We consider the situation where we observe a set of random variables satisfying the so-called back-door criterion. When the ordinary least squares method is utilized to estimate the total effect, we formulate the unbiased estimator of the causal effect (the estimated causal effect) on the variance of the outcome variable with external intervention in which a treatment variable is set to a specified constant value. In addition, we provide the variance formula for the estimated causal effect on the variance. The variance formula proposed in this paper is exact, in contrast to those in most previous studies on estimating causal effects.

本論文では，確率変数間の因果関係（データ生成過程）が線形構造方程式モデルと対応する非巡回的有向グラフにより記述できる状況を考える．未観測交絡因子が存在する場合，観測された相関情報から総合効果を推定する方法論として，操作変数法とフロントドア基準に基づいた二段階最小二乗法がある．しかし，これらの2つの推定方法のいずれもが適用できる場合において，総合効果の推定精度の観点では，これら2つのあいだに定性的な優劣関係を与えることはできない．そこで，本論文では，操作変数法とフロントドア基準によって推定された総合効果を統合した新たな推定量を提案する．また，数値実験をとおして，総合効果の推定精度の観点から，(i)この統合型推定量が個々の推定量よりも優れていること，そして，(ii)仮に未観測交絡因子がなく，通常最小二乗推定量（バックドア基準）が個々の推定量よりも優れている場合であっても，統合型推定量のほうが通常最小二乗推定量よりも優れているケースがあることを指摘する．